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<title>Factorization A: Orbital Case</title>
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    <pre><span style="font-style: normal; font-weight: 700"><font color="#0000FF" size="6">FACTORIZATION OF A</font></span></pre>
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    <pre><span style="font-style: normal; font-weight: 700"><font color="#FF0000" size="4">and</font></span></pre>
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    <pre><span style="font-style: normal; font-weight: 700"><font size="6" color="#0000FF">Phase Advance Computation </font></span></pre>
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<address align="left"><font color="#9900FF" size="6"><b>Frank Schmidt wants me 
  to report on BS: click <a href="#bs">here</a></b></font></address>
<p><font color="#FF0000"><b>M denotes a DAMAP and DS denotes a DAMAPSPIN:&nbsp;&nbsp;&nbsp; 
DS=(M,S)&nbsp; where S is a 3 by 3 matrix.</b></font></p>
<p><font color="#FF0000"><b>A symplectic map M can be put in normal form</b></font></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
R=A<sup>-1 </sup>* M * A</p>
<p><font color="#FF0000"><b>FPP&nbsp; does choose an A very carefully. If one 
insists on a &quot;canonical&quot; form for A, then the following routine can be called.</b></font></p>
<font SIZE="2">
<p></font><span style="text-transform: uppercase">
<font SIZE="2" COLOR="#0000ff"><b>call</b></font></span><font SIZE="2"><span style="text-transform: uppercase"> 
factor(<font color="#0000FF">a_t,a_f,a_l,a_nl</font>,<font color="#FF0000">&nbsp; DR=dr</font>,<font color="#FF00FF">&nbsp; r_te=r_te,cs_te=cs_te,cosLIKE=cosLIKE</font>)</span></p>
</font>
<p><span style="text-transform: uppercase"><font color="#0000FF" size="6">
<b>1)</b></font><font color="#0000FF" size="5"> </font>
<font color="#0000FF" SIZE="2">
&nbsp;a_t=a_f * a_l * a_nl </font></span><font color="#0000FF" SIZE="2">
: These are compulsory DAMAPS. </font></p>

<p><font size="2" color="#0000FF">A_F=Parameter dependent fixed point map; may 
include time in coasting beam case ndpt/=0.</font></p>

<p><font size="2" color="#0000FF">A_L= Parameter dependent linear map around the 
fixed point!</font></p>

<p><font size="2" color="#0000FF">A_NL = the nonlinear part of A around the 
fixed point!.</font></p>

<p><font color="#0000FF" size="6"><b>2)</b> </font>
<font size="2" color="#0000FF">&nbsp;Difference between the input A_T and its 
value on exit, i.e., puts A_T in &quot;canonical&quot; form. </font></p>

<p><font size="2" color="#0000FF">A_T<sub>entrance</sub> = A_T<sub>exit</sub> *
</font><font size="2" color="#FF0000"><b>DR</b></font><font size="2" color="#0000FF"> 
; </font><font size="2" color="#FF0000"><b>DR</b></font><font size="2" color="#0000FF"> 
is the phase advance (nonlinear terms included). Notice that A_L and A_NL are 
changed by </font><b><font size="2" color="#FF0000">DR</font></b><font size="2" color="#0000FF">.</font></p>

<p><u>Definition of the canonical form of A:&nbsp; </u></p>
<ul>
  <li><font size="2">Linear part: A<sub>12</sub>=A<sub>34</sub>(=A<sub>56</sub>)=0
  <font face="Times New Roman">&nbsp;&#8594; that is a Langrangian definition of 
  A consistent with Courant-Snyder. </font></font>
  <font face="Times New Roman" size="2">It insures that the phase advance 
  between monitors is the phase difference of the position oscillations. This 
  concept has no easy extension to the nonlinear case unfortunately.</font></li>
  <li><font face="Times New Roman" size="2">In the nonlinear case, FPP offers 
  two choices:</font></li>
</ul>
<blockquote>
  <ol>
    <li><font face="Times New Roman" size="2">The reverse Dragt-Finn 
    representation has no tune shift phasors&nbsp; </font><font size="2">&nbsp;<font face="Times New Roman"> 
    &#8594; </font></font><font face="Times New Roman" size="2">&nbsp; </font>
    <font SIZE="2" color="#008000"><b>onelie=false</b></font></li>
    <li><font face="Times New Roman" size="2">The one-lie exponent 
    representation has no tune shift phasors&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
    </font><font size="2">&nbsp;<font face="Times New Roman"> &#8594; </font></font>
    <font face="Times New Roman" size="2">&nbsp; </font>
    <font SIZE="2" color="#008000"><b>onelie=true</b></font></li>
  </ol>
</blockquote>
<p><font color="#FF0000"><b>DR</b></font> contains the phase difference between 
the input map and its canonical form</p>
<p><b><font color="#0000FF" size="6">3)</font></b><font size="7" color="#0000FF">
</font>Some people like this  parameterization of Teng and Edwards for a 4x4 
oscillatory matrix. (No RF-Cavity). </p>
<p><font SIZE="2"><span style="text-transform: uppercase"><font color="#0000FF">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
a_l=</font><font color="#FF00FF"> R_TE * cs_te</font></span></font></p>
<p><img border="0" src="factor2.gif" width="536" height="92"></p>
<p><img border="0" src="factor1.gif" width="272" height="75"><img border="0" src="factor6.gif" width="475" height="65"></p>
<p><font size="2">The parameter <span style="text-transform: uppercase"><font color="#FF00FF">&nbsp;<b>cosLIKE</b> </font></span>
is </font></p>
<ul>
  <li><font size="2"><b><font color="#0000FF">TRUE</font></b> if the component c and s of <font color="#FF00FF">R_TE</font>&nbsp;&nbsp; 
  are&nbsp; </font>
  <b><font color="#FF0000" size="4">c=cos(<font face="Symbol">j</font>)&nbsp; and s=sin(<font face="Symbol">j</font>)
  </font></b>
  </li>
  <li><font size="2"><b><font color="#0000FF">FALSE</font></b> if the component c and s of <font color="#FF00FF">
  R_TE</font>&nbsp;&nbsp; are&nbsp;</font><b><font color="#FF0000" size="4"> c=cosh(<font face="Symbol">j</font>)&nbsp; 
  and s=sinh(<font face="Symbol">j</font>) </font></b><font size="2">with -s on the second row also equal 
  to sinh(<font face="Symbol">j</font>).</font></li>
</ul>
<p>&nbsp;</p>
<p><u>Example routine from PTC </u><a href="twiss.htm">Click here</a></p>
<hr>
<p><u><b><font color="#FF00FF">BS Report:</font></b></u></p>
<p><font size="2"><a name="bs"></a><font color="#FF00FF">Apparently it is 
unknown, according to Frank, that the coefficient &quot;c&quot; of the matrix <span style="text-transform: uppercase">
R_TE </span>can be greater than 1! This is actually implicit in the paper of 
Ohmi, Hirata and Oide of 1994. However Frank may well be correct. According to 
Ohmi, the reviewer of a recent paper by Y. Seimiya, K. Ohmi, D. Zhou, J. W. 
Flanagan and Y. Ohnishi needed to be reminded that &quot;c&quot; can be greater than one. 
It is in reference [7] of that paper. Nothing new to competent people 
apparently!</font></font></p>
<p>&nbsp;</p>
<p>&nbsp;</p>

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